Construct bijections $f_1 : ((0,1)times(2,3)) rightarrow (0,2)times(5,6)$ and $f_2 :...
I need to construct two bijections:
$$f_1 : ((0,1)times(2,3)) rightarrow (0,2)times(5,6)$$
$$f_2 : (mathbb{Z}times[0,1))rightarrowmathbb{R}$$
I know what bijection means and all conditions that functions have to fulfill in order to be bijective, but I have no idea how should I 'construct' them. I thought of drawing graphs of each set from function $f_1$, but it does not help me to do further steps.
It would be nice if you could show me step-by-step how it should be done.
elementary-set-theory
asked Dec 8 at 15:48
whiskeyo
536
|
show 3 more comments
I need to construct two bijections:
$$f_1 : ((0,1)times(2,3)) rightarrow (0,2)times(5,6)$$
$$f_2 : (mathbb{Z}times[0,1))rightarrowmathbb{R}$$
I know what bijection means and all conditions that functions have to fulfill in order to be bijective, but I have no idea how should I 'construct' them. I thought of drawing graphs of each set from function $f_1$, but it does not help me to do further steps.
It would be nice if you could show me step-by-step how it should be done.
elementary-set-theory
asked Dec 8 at 15:48
whiskeyo
536
Don't get too hung up on the word "construct". If you were just asked to find the bijections, would you know what to do?
– Henning Makholm
Dec 8 at 15:57
I know that I should somehow find functions which fit these sets, but how can I do it if not by guessing?
– whiskeyo
Dec 8 at 15:59
Have you tried just "guessing"? This is not a follow-a- method exercise, it's just a check question to make sure you have understood what a bijection is.
– Henning Makholm
Dec 8 at 16:06
I tried to transform $f_1 : ((0,1)times(2,3)) rightarrow (0,2)times(5,6)$ into $f_1 : A rightarrow B$, where $A : (0,1)rightarrow (2,3)$ and $B : (0,2)rightarrow (5,6)$ and then find functions fitting both sets, but I could not do that.
– whiskeyo
Dec 8 at 16:23
Hmmm. Can you make just a function that maps (0,1) bijectively to (2,3)?
– Henning Makholm
Dec 8 at 16:33
|
show 3 more comments
I need to construct two bijections:
$$f_1 : ((0,1)times(2,3)) rightarrow (0,2)times(5,6)$$
$$f_2 : (mathbb{Z}times[0,1))rightarrowmathbb{R}$$
I know what bijection means and all conditions that functions have to fulfill in order to be bijective, but I have no idea how should I 'construct' them. I thought of drawing graphs of each set from function $f_1$, but it does not help me to do further steps.
It would be nice if you could show me step-by-step how it should be done.
elementary-set-theory
asked Dec 8 at 15:48
whiskeyo
536
I need to construct two bijections:
$$f_1 : ((0,1)times(2,3)) rightarrow (0,2)times(5,6)$$
$$f_2 : (mathbb{Z}times[0,1))rightarrowmathbb{R}$$
I know what bijection means and all conditions that functions have to fulfill in order to be bijective, but I have no idea how should I 'construct' them. I thought of drawing graphs of each set from function $f_1$, but it does not help me to do further steps.
It would be nice if you could show me step-by-step how it should be done.
elementary-set-theory
elementary-set-theory
asked Dec 8 at 15:48
whiskeyo
536
asked Dec 8 at 15:48
whiskeyo
536
edited Dec 8 at 18:24
asked Dec 8 at 15:48
whiskeyo
536
asked Dec 8 at 15:48
whiskeyo
536
asked Dec 8 at 15:48
whiskeyo
536
536
Don't get too hung up on the word "construct". If you were just asked to find the bijections, would you know what to do?
– Henning Makholm
Dec 8 at 15:57
I know that I should somehow find functions which fit these sets, but how can I do it if not by guessing?
– whiskeyo
Dec 8 at 15:59
Have you tried just "guessing"? This is not a follow-a- method exercise, it's just a check question to make sure you have understood what a bijection is.
– Henning Makholm
Dec 8 at 16:06
I tried to transform $f_1 : ((0,1)times(2,3)) rightarrow (0,2)times(5,6)$ into $f_1 : A rightarrow B$, where $A : (0,1)rightarrow (2,3)$ and $B : (0,2)rightarrow (5,6)$ and then find functions fitting both sets, but I could not do that.
– whiskeyo
Dec 8 at 16:23
Hmmm. Can you make just a function that maps (0,1) bijectively to (2,3)?
– Henning Makholm
Dec 8 at 16:33
|
show 3 more comments
Don't get too hung up on the word "construct". If you were just asked to find the bijections, would you know what to do?
– Henning Makholm
Dec 8 at 15:57
I know that I should somehow find functions which fit these sets, but how can I do it if not by guessing?
– whiskeyo
Dec 8 at 15:59
Have you tried just "guessing"? This is not a follow-a- method exercise, it's just a check question to make sure you have understood what a bijection is.
– Henning Makholm
Dec 8 at 16:06
I tried to transform $f_1 : ((0,1)times(2,3)) rightarrow (0,2)times(5,6)$ into $f_1 : A rightarrow B$, where $A : (0,1)rightarrow (2,3)$ and $B : (0,2)rightarrow (5,6)$ and then find functions fitting both sets, but I could not do that.
– whiskeyo
Dec 8 at 16:23
Hmmm. Can you make just a function that maps (0,1) bijectively to (2,3)?
– Henning Makholm
Dec 8 at 16:33
Don't get too hung up on the word "construct". If you were just asked to find the bijections, would you know what to do?
– Henning Makholm
Dec 8 at 15:57
Don't get too hung up on the word "construct". If you were just asked to find the bijections, would you know what to do?
– Henning Makholm
Dec 8 at 15:57
I know that I should somehow find functions which fit these sets, but how can I do it if not by guessing?
– whiskeyo
Dec 8 at 15:59
I know that I should somehow find functions which fit these sets, but how can I do it if not by guessing?
– whiskeyo
Dec 8 at 15:59
Have you tried just "guessing"? This is not a follow-a- method exercise, it's just a check question to make sure you have understood what a bijection is.
– Henning Makholm
Dec 8 at 16:06
Have you tried just "guessing"? This is not a follow-a- method exercise, it's just a check question to make sure you have understood what a bijection is.
– Henning Makholm
Dec 8 at 16:06
I tried to transform $f_1 : ((0,1)times(2,3)) rightarrow (0,2)times(5,6)$ into $f_1 : A rightarrow B$, where $A : (0,1)rightarrow (2,3)$ and $B : (0,2)rightarrow (5,6)$ and then find functions fitting both sets, but I could not do that.
– whiskeyo
Dec 8 at 16:23
I tried to transform $f_1 : ((0,1)times(2,3)) rightarrow (0,2)times(5,6)$ into $f_1 : A rightarrow B$, where $A : (0,1)rightarrow (2,3)$ and $B : (0,2)rightarrow (5,6)$ and then find functions fitting both sets, but I could not do that.
– whiskeyo
Dec 8 at 16:23
Hmmm. Can you make just a function that maps (0,1) bijectively to (2,3)?
– Henning Makholm
Dec 8 at 16:33
Hmmm. Can you make just a function that maps (0,1) bijectively to (2,3)?
– Henning Makholm
Dec 8 at 16:33
|
show 3 more comments
1 Answer
1
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oldest
votes
For $f_2$,The usual euclidean product of sets may be a little confusing here.
Try thinking of "$mathbb{Z}times[0,1)$" as the set "To each integer, assign an interval from 0 to 1." Then $f_2(x,t) = x+t$ is a bijection to $mathbb{R}$, where $xinmathbb{Z}$ is the integer part of some real number, and $tin[0,1)$ is the decimal part of that number.
answered Dec 8 at 18:37
Adam Cartisano
1614
add a comment |
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1 Answer
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1 Answer
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active
oldest
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active
oldest
votes
For $f_2$,The usual euclidean product of sets may be a little confusing here.
Try thinking of "$mathbb{Z}times[0,1)$" as the set "To each integer, assign an interval from 0 to 1." Then $f_2(x,t) = x+t$ is a bijection to $mathbb{R}$, where $xinmathbb{Z}$ is the integer part of some real number, and $tin[0,1)$ is the decimal part of that number.
answered Dec 8 at 18:37
Adam Cartisano
1614
add a comment |
For $f_2$,The usual euclidean product of sets may be a little confusing here.
Try thinking of "$mathbb{Z}times[0,1)$" as the set "To each integer, assign an interval from 0 to 1." Then $f_2(x,t) = x+t$ is a bijection to $mathbb{R}$, where $xinmathbb{Z}$ is the integer part of some real number, and $tin[0,1)$ is the decimal part of that number.
answered Dec 8 at 18:37
Adam Cartisano
1614
add a comment |
For $f_2$,The usual euclidean product of sets may be a little confusing here.
Try thinking of "$mathbb{Z}times[0,1)$" as the set "To each integer, assign an interval from 0 to 1." Then $f_2(x,t) = x+t$ is a bijection to $mathbb{R}$, where $xinmathbb{Z}$ is the integer part of some real number, and $tin[0,1)$ is the decimal part of that number.
answered Dec 8 at 18:37
Adam Cartisano
1614
For $f_2$,The usual euclidean product of sets may be a little confusing here.
Try thinking of "$mathbb{Z}times[0,1)$" as the set "To each integer, assign an interval from 0 to 1." Then $f_2(x,t) = x+t$ is a bijection to $mathbb{R}$, where $xinmathbb{Z}$ is the integer part of some real number, and $tin[0,1)$ is the decimal part of that number.
answered Dec 8 at 18:37
Adam Cartisano
1614
answered Dec 8 at 18:37
Adam Cartisano
1614
answered Dec 8 at 18:37
Adam Cartisano
1614
answered Dec 8 at 18:37
Adam Cartisano
1614
1614
add a comment |
add a comment |
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Don't get too hung up on the word "construct". If you were just asked to find the bijections, would you know what to do?
– Henning Makholm
Dec 8 at 15:57
I know that I should somehow find functions which fit these sets, but how can I do it if not by guessing?
– whiskeyo
Dec 8 at 15:59
Have you tried just "guessing"? This is not a follow-a- method exercise, it's just a check question to make sure you have understood what a bijection is.
– Henning Makholm
Dec 8 at 16:06
I tried to transform $f_1 : ((0,1)times(2,3)) rightarrow (0,2)times(5,6)$ into $f_1 : A rightarrow B$, where $A : (0,1)rightarrow (2,3)$ and $B : (0,2)rightarrow (5,6)$ and then find functions fitting both sets, but I could not do that.
– whiskeyo
Dec 8 at 16:23
Hmmm. Can you make just a function that maps (0,1) bijectively to (2,3)?
– Henning Makholm
Dec 8 at 16:33